SQ 012

Shared Qs (012)

Question

A list of inputs and outputs is expressed as \((x,y)\) pairs.

( 8 , 2 )

( 7 , 4 )

( 1 , 3 )

( 5 , 6 )

( 2 , 5 )

( 6 , 2 )

( 6 , 7 )

Is this list consistent with \(y\) being a function of \(x\)?

Is this list consistent with \(x\) being a function of \(y\)?

Is this list consistent with a one-to-one function?
Solution

An \(x\) value repeats with different \(y\) values, so \(y\) is not a function of \(x\). For example: (6, 2) and (6, 7)

Also, a \(y\) value repeates with different \(x\) values, so \(x\) is not a function of \(y\). For example: (8, 2) and (6, 2)

We would need both to be function of the other to have a one-to-one function. Neither is, so we don’t even have a function in either direction.
Question

A list of inputs and outputs is expressed as \((x,y)\) pairs.

( 1 , 3 )

( 4 , 1 )

( 2 , 9 )

( 5 , 4 )

( 6 , 5 )

( 6 , 5 )

Is this list consistent with \(y\) being a function of \(x\)?

Is this list consistent with \(x\) being a function of \(y\)?

Is this list consistent with a one-to-one function?
Solution

Every repeated \(x\) value is matched with a consistent (also repeating) \(y\), and vice versa. This means \(y\) is a function of \(x\) and \(x\) is a function of \(y\), and the list is consistent with a one-to-one function.
Question

A list of inputs and outputs is expressed as \((x,y)\) pairs.

( 8 , 7 )

( 4 , 5 )

( 3 , 3 )

( 4 , 5 )

( 1 , 5 )

Is this list consistent with \(y\) being a function of \(x\)?

Is this list consistent with \(x\) being a function of \(y\)?

Is this list consistent with a one-to-one function?
Solution

Every repeated \(x\) value is matched with a consistent (also repeating) \(y\), so \(y\) is a function of \(x\).

A \(y\) value repeates with different \(x\) values, so \(x\) is not a function of \(y\). For example: (4, 5) and (1, 5)

Because \(x\) is not a function of \(y\), this list is inconsistent with a one-to-one function.
Question

A list of inputs and outputs is expressed as \((x,y)\) pairs.

( 5 , 1 )

( 6 , 8 )

( 3 , 4 )

( 1 , 4 )

( 1 , 4 )

Is this list consistent with \(y\) being a function of \(x\)?

Is this list consistent with \(x\) being a function of \(y\)?

Is this list consistent with a one-to-one function?
Solution

Every repeated \(x\) value is matched with a consistent (also repeating) \(y\), so \(y\) is a function of \(x\).

A \(y\) value repeates with different \(x\) values, so \(x\) is not a function of \(y\). For example: (3, 4) and (1, 4)

Because \(x\) is not a function of \(y\), this list is inconsistent with a one-to-one function.
Question

A list of inputs and outputs is expressed as \((x,y)\) pairs.

( 9 , 4 )

( 6 , 5 )

( 6 , 5 )

( 9 , 4 )

( 4 , 2 )

Is this list consistent with \(y\) being a function of \(x\)?

Is this list consistent with \(x\) being a function of \(y\)?

Is this list consistent with a one-to-one function?
Solution

Every repeated \(x\) value is matched with a consistent (also repeating) \(y\), and vice versa. This means \(y\) is a function of \(x\) and \(x\) is a function of \(y\), and the list is consistent with a one-to-one function.
Question

A list of inputs and outputs is expressed as \((x,y)\) pairs.

( 4 , 7 )

( 9 , 9 )

( 5 , 8 )

( 7 , 3 )

( 8 , 5 )

( 2 , 1 )

( 5 , 8 )

Is this list consistent with \(y\) being a function of \(x\)?

Is this list consistent with \(x\) being a function of \(y\)?

Is this list consistent with a one-to-one function?
Solution

Every repeated \(x\) value is matched with a consistent (also repeating) \(y\), and vice versa. This means \(y\) is a function of \(x\) and \(x\) is a function of \(y\), and the list is consistent with a one-to-one function.
Question

A list of inputs and outputs is expressed as \((x,y)\) pairs.

( 3 , 2 )

( 5 , 6 )

( 2 , 4 )

( 4 , 9 )

( 8 , 3 )

( 8 , 3 )

Is this list consistent with \(y\) being a function of \(x\)?

Is this list consistent with \(x\) being a function of \(y\)?

Is this list consistent with a one-to-one function?
Solution

Every repeated \(x\) value is matched with a consistent (also repeating) \(y\), and vice versa. This means \(y\) is a function of \(x\) and \(x\) is a function of \(y\), and the list is consistent with a one-to-one function.
Question

A list of inputs and outputs is expressed as \((x,y)\) pairs.

( 5 , 4 )

( 2 , 1 )

( 6 , 6 )

( 6 , 4 )

( 8 , 8 )

( 8 , 7 )

Is this list consistent with \(y\) being a function of \(x\)?

Is this list consistent with \(x\) being a function of \(y\)?

Is this list consistent with a one-to-one function?
Solution

An \(x\) value repeats with different \(y\) values, so \(y\) is not a function of \(x\). For example: (8, 8) and (8, 7)

Also, a \(y\) value repeates with different \(x\) values, so \(x\) is not a function of \(y\). For example: (5, 4) and (6, 4)

We would need both to be function of the other to have a one-to-one function. Neither is, so we don’t even have a function in either direction.
Question

A list of inputs and outputs is expressed as \((x,y)\) pairs.

( 7 , 3 )

( 3 , 8 )

( 5 , 6 )

( 1 , 6 )

( 1 , 9 )

Is this list consistent with \(y\) being a function of \(x\)?

Is this list consistent with \(x\) being a function of \(y\)?

Is this list consistent with a one-to-one function?
Solution

An \(x\) value repeats with different \(y\) values, so \(y\) is not a function of \(x\). For example: (1, 6) and (1, 9)

Also, a \(y\) value repeates with different \(x\) values, so \(x\) is not a function of \(y\). For example: (5, 6) and (1, 6)

We would need both to be function of the other to have a one-to-one function. Neither is, so we don’t even have a function in either direction.
Question

A list of inputs and outputs is expressed as \((x,y)\) pairs.

( 3 , 3 )

( 8 , 7 )

( 1 , 9 )

( 4 , 5 )

( 1 , 9 )

( 7 , 6 )

( 1 , 9 )

Is this list consistent with \(y\) being a function of \(x\)?

Is this list consistent with \(x\) being a function of \(y\)?

Is this list consistent with a one-to-one function?
Solution

Every repeated \(x\) value is matched with a consistent (also repeating) \(y\), and vice versa. This means \(y\) is a function of \(x\) and \(x\) is a function of \(y\), and the list is consistent with a one-to-one function.
Question

Are these points consistent with \(y\) being a function of \(x\)?

Are these points consistent with \(x\) being a function of \(y\)?

Are these points consistent with a one-to-one function?
Solution

It is best to visualize the vertical line test and the horizontal line test. If you can draw a vertical line through 2 (or more) points, then \(y\) is not a function of \(x\). If you can draw a horizontal line through 2 (or more) points, then \(x\) is not a function of \(y\).

Every \(x\) value has only one \(y\) value, so \(y\) is a function of \(x\). The graph passes the vertical line test.

A \(y\) value repeates with different \(x\) values, so \(x\) is not a function of \(y\). The graph fails the horizontal line test. For example: (8, 3) and (6, 3)

Because \(x\) is not a function of \(y\), these points are inconsistent with a one-to-one function.
Question

Are these points consistent with \(y\) being a function of \(x\)?

Are these points consistent with \(x\) being a function of \(y\)?

Are these points consistent with a one-to-one function?
Solution

It is best to visualize the vertical line test and the horizontal line test. If you can draw a vertical line through 2 (or more) points, then \(y\) is not a function of \(x\). If you can draw a horizontal line through 2 (or more) points, then \(x\) is not a function of \(y\).

An \(x\) value repeats with different \(y\) values, so \(y\) is not a function of \(x\). The graph fails the vertical-line test. For example: (10, 1) and (10, 8)

Every \(y\) value has only one \(x\) value, so \(x\) is a function of \(y\). The graph passes the horizontal line test.

Because \(y\) is not a function of \(x\), these points are inconsistent with a one-to-one function.
Question

Are these points consistent with \(y\) being a function of \(x\)?

Are these points consistent with \(x\) being a function of \(y\)?

Are these points consistent with a one-to-one function?
Solution

It is best to visualize the vertical line test and the horizontal line test. If you can draw a vertical line through 2 (or more) points, then \(y\) is not a function of \(x\). If you can draw a horizontal line through 2 (or more) points, then \(x\) is not a function of \(y\).

Every \(x\) value has only one \(y\) value, so \(y\) is a function of \(x\). The graph passes the vertical line test.

A \(y\) value repeates with different \(x\) values, so \(x\) is not a function of \(y\). The graph fails the horizontal line test. For example: (10, 1) and (9, 1)

Because \(x\) is not a function of \(y\), these points are inconsistent with a one-to-one function.
Question

Are these points consistent with \(y\) being a function of \(x\)?

Are these points consistent with \(x\) being a function of \(y\)?

Are these points consistent with a one-to-one function?
Solution

It is best to visualize the vertical line test and the horizontal line test. If you can draw a vertical line through 2 (or more) points, then \(y\) is not a function of \(x\). If you can draw a horizontal line through 2 (or more) points, then \(x\) is not a function of \(y\).

Every \(x\) value has only one \(y\) value, and vice versa. The graph passes both the vertical-line and horizontal-line tests. This means \(y\) is a function of \(x\) and \(x\) is a function of \(y\), and the points are consistent with a one-to-one function.
Question

Are these points consistent with \(y\) being a function of \(x\)?

Are these points consistent with \(x\) being a function of \(y\)?

Are these points consistent with a one-to-one function?
Solution

It is best to visualize the vertical line test and the horizontal line test. If you can draw a vertical line through 2 (or more) points, then \(y\) is not a function of \(x\). If you can draw a horizontal line through 2 (or more) points, then \(x\) is not a function of \(y\).

An \(x\) value repeats with different \(y\) values, so \(y\) is not a function of \(x\). The graph fails the vertical line test. For example: (4, 9) and (4, 3)

Also, a \(y\) value repeates with different \(x\) values, so \(x\) is not a function of \(y\). The graph fails the horizontal line test. For example: (3, 9) and (4, 9)

We would need both to be function of the other to have a one-to-one function. Neither is, so we don’t even have a function in either direction.
Question

Are these points consistent with \(y\) being a function of \(x\)?

Are these points consistent with \(x\) being a function of \(y\)?

Are these points consistent with a one-to-one function?
Solution

It is best to visualize the vertical line test and the horizontal line test. If you can draw a vertical line through 2 (or more) points, then \(y\) is not a function of \(x\). If you can draw a horizontal line through 2 (or more) points, then \(x\) is not a function of \(y\).

Every \(x\) value has only one \(y\) value, and vice versa. The graph passes both the vertical-line and horizontal-line tests. This means \(y\) is a function of \(x\) and \(x\) is a function of \(y\), and the points are consistent with a one-to-one function.
Question

Are these points consistent with \(y\) being a function of \(x\)?

Are these points consistent with \(x\) being a function of \(y\)?

Are these points consistent with a one-to-one function?
Solution

It is best to visualize the vertical line test and the horizontal line test. If you can draw a vertical line through 2 (or more) points, then \(y\) is not a function of \(x\). If you can draw a horizontal line through 2 (or more) points, then \(x\) is not a function of \(y\).

Every \(x\) value has only one \(y\) value, and vice versa. The graph passes both the vertical-line and horizontal-line tests. This means \(y\) is a function of \(x\) and \(x\) is a function of \(y\), and the points are consistent with a one-to-one function.
Question

Are these points consistent with \(y\) being a function of \(x\)?

Are these points consistent with \(x\) being a function of \(y\)?

Are these points consistent with a one-to-one function?
Solution

It is best to visualize the vertical line test and the horizontal line test. If you can draw a vertical line through 2 (or more) points, then \(y\) is not a function of \(x\). If you can draw a horizontal line through 2 (or more) points, then \(x\) is not a function of \(y\).

Every \(x\) value has only one \(y\) value, so \(y\) is a function of \(x\). The graph passes the vertical line test.

A \(y\) value repeates with different \(x\) values, so \(x\) is not a function of \(y\). The graph fails the horizontal line test. For example: (5, 6) and (6, 6)

Because \(x\) is not a function of \(y\), these points are inconsistent with a one-to-one function.
Question

Are these points consistent with \(y\) being a function of \(x\)?

Are these points consistent with \(x\) being a function of \(y\)?

Are these points consistent with a one-to-one function?
Solution

It is best to visualize the vertical line test and the horizontal line test. If you can draw a vertical line through 2 (or more) points, then \(y\) is not a function of \(x\). If you can draw a horizontal line through 2 (or more) points, then \(x\) is not a function of \(y\).

Every \(x\) value has only one \(y\) value, so \(y\) is a function of \(x\). The graph passes the vertical line test.

A \(y\) value repeates with different \(x\) values, so \(x\) is not a function of \(y\). The graph fails the horizontal line test. For example: (1, 5) and (7, 5)

Because \(x\) is not a function of \(y\), these points are inconsistent with a one-to-one function.
Question

Are these points consistent with \(y\) being a function of \(x\)?

Are these points consistent with \(x\) being a function of \(y\)?

Are these points consistent with a one-to-one function?
Solution

It is best to visualize the vertical line test and the horizontal line test. If you can draw a vertical line through 2 (or more) points, then \(y\) is not a function of \(x\). If you can draw a horizontal line through 2 (or more) points, then \(x\) is not a function of \(y\).

Every \(x\) value has only one \(y\) value, so \(y\) is a function of \(x\). The graph passes the vertical line test.

A \(y\) value repeates with different \(x\) values, so \(x\) is not a function of \(y\). The graph fails the horizontal line test. For example: (3, 4) and (10, 4)

Because \(x\) is not a function of \(y\), these points are inconsistent with a one-to-one function.
Question

Are these connections consistent with \(y\) being a function of \(x\)?

Are these connections consistent with \(x\) being a function of \(y\)?

Are these connections consistent with a one-to-one function?
Solution

If any value of \(x\) has two different connections, then \(y\) is not a function of \(x\). If any value of \(y\) has two different connections, then \(x\) is not a function of \(y\).

Every \(x\) value has only one connection, and vice versa. This means \(y\) is a function of \(x\) and \(x\) is a function of \(y\), and the connections are consistent with a one-to-one function.
Question

Are these connections consistent with \(y\) being a function of \(x\)?

Are these connections consistent with \(x\) being a function of \(y\)?

Are these connections consistent with a one-to-one function?
Solution

If any value of \(x\) has two different connections, then \(y\) is not a function of \(x\). If any value of \(y\) has two different connections, then \(x\) is not a function of \(y\).

An \(x\) value has two connections, so \(y\) is not a function of \(x\). For example: (2, 6) and (2, 1)

Every \(y\) value has only one connection, so \(x\) is a function of \(y\).

Because \(y\) is not a function of \(x\), these connections are inconsistent with a one-to-one function.
Question

Are these connections consistent with \(y\) being a function of \(x\)?

Are these connections consistent with \(x\) being a function of \(y\)?

Are these connections consistent with a one-to-one function?
Solution

If any value of \(x\) has two different connections, then \(y\) is not a function of \(x\). If any value of \(y\) has two different connections, then \(x\) is not a function of \(y\).

Every \(x\) value has only one connection, and vice versa. This means \(y\) is a function of \(x\) and \(x\) is a function of \(y\), and the connections are consistent with a one-to-one function.
Question

Are these connections consistent with \(y\) being a function of \(x\)?

Are these connections consistent with \(x\) being a function of \(y\)?

Are these connections consistent with a one-to-one function?
Solution

If any value of \(x\) has two different connections, then \(y\) is not a function of \(x\). If any value of \(y\) has two different connections, then \(x\) is not a function of \(y\).

Every \(x\) value has only one connection, and vice versa. This means \(y\) is a function of \(x\) and \(x\) is a function of \(y\), and the connections are consistent with a one-to-one function.
Question

Are these connections consistent with \(y\) being a function of \(x\)?

Are these connections consistent with \(x\) being a function of \(y\)?

Are these connections consistent with a one-to-one function?
Solution

If any value of \(x\) has two different connections, then \(y\) is not a function of \(x\). If any value of \(y\) has two different connections, then \(x\) is not a function of \(y\).

Every \(x\) value has only one connection, so \(y\) is a function of \(x\).

A \(y\) value has two connections, so \(x\) is not a function of \(y\). For example: (0, 2) and (1, 2)

Because \(x\) is not a function of \(y\), these connections are inconsistent with a one-to-one function.
Question

Are these connections consistent with \(y\) being a function of \(x\)?

Are these connections consistent with \(x\) being a function of \(y\)?

Are these connections consistent with a one-to-one function?
Solution

If any value of \(x\) has two different connections, then \(y\) is not a function of \(x\). If any value of \(y\) has two different connections, then \(x\) is not a function of \(y\).

An \(x\) value has two different connections, so \(y\) is not a function of \(x\). For example: (4, 5) and (4, 0)

Also, a \(y\) value has two different connections, so \(x\) is not a function of \(y\). For example: (5, 4) and (7, 4)

We would need both to be function of the other to have a one-to-one function. Neither is, so we don’t even have a function in either direction.
Question

Are these connections consistent with \(y\) being a function of \(x\)?

Are these connections consistent with \(x\) being a function of \(y\)?

Are these connections consistent with a one-to-one function?
Solution

If any value of \(x\) has two different connections, then \(y\) is not a function of \(x\). If any value of \(y\) has two different connections, then \(x\) is not a function of \(y\).

An \(x\) value has two connections, so \(y\) is not a function of \(x\). For example: (4, 2) and (4, 8)

Every \(y\) value has only one connection, so \(x\) is a function of \(y\).

Because \(y\) is not a function of \(x\), these connections are inconsistent with a one-to-one function.
Question

Are these connections consistent with \(y\) being a function of \(x\)?

Are these connections consistent with \(x\) being a function of \(y\)?

Are these connections consistent with a one-to-one function?
Solution

If any value of \(x\) has two different connections, then \(y\) is not a function of \(x\). If any value of \(y\) has two different connections, then \(x\) is not a function of \(y\).

Every \(x\) value has only one connection, and vice versa. This means \(y\) is a function of \(x\) and \(x\) is a function of \(y\), and the connections are consistent with a one-to-one function.
Question

Are these connections consistent with \(y\) being a function of \(x\)?

Are these connections consistent with \(x\) being a function of \(y\)?

Are these connections consistent with a one-to-one function?
Solution

If any value of \(x\) has two different connections, then \(y\) is not a function of \(x\). If any value of \(y\) has two different connections, then \(x\) is not a function of \(y\).

An \(x\) value has two connections, so \(y\) is not a function of \(x\). For example: (2, 2) and (2, 3)

Every \(y\) value has only one connection, so \(x\) is a function of \(y\).

Because \(y\) is not a function of \(x\), these connections are inconsistent with a one-to-one function.
Question

Are these connections consistent with \(y\) being a function of \(x\)?

Are these connections consistent with \(x\) being a function of \(y\)?

Are these connections consistent with a one-to-one function?
Solution

If any value of \(x\) has two different connections, then \(y\) is not a function of \(x\). If any value of \(y\) has two different connections, then \(x\) is not a function of \(y\).

An \(x\) value has two connections, so \(y\) is not a function of \(x\). For example: (2, 9) and (2, 6)

Every \(y\) value has only one connection, so \(x\) is a function of \(y\).

Because \(y\) is not a function of \(x\), these connections are inconsistent with a one-to-one function.
Question

Is this curve consistent with \(y\) being a function of \(x\)?

Is this curve consistent with \(x\) being a function of \(y\)?

Is this curve consistent with a one-to-one function?
Solution

It is best to visualize the vertical line test and the horizontal line test. If you can draw a vertical line through 2 (or more) points, then \(y\) is not a function of \(x\). If you can draw a horizontal line through 2 (or more) points, then \(x\) is not a function of \(y\).

The function is one-to-one if and only if \(y\) is a function of \(x\) and \(x\) is a function of \(y\).

Below, if the function is not one-to-one, some failed line tests are shown:
Question

Is this curve consistent with \(y\) being a function of \(x\)?

Is this curve consistent with \(x\) being a function of \(y\)?

Is this curve consistent with a one-to-one function?
Solution

It is best to visualize the vertical line test and the horizontal line test. If you can draw a vertical line through 2 (or more) points, then \(y\) is not a function of \(x\). If you can draw a horizontal line through 2 (or more) points, then \(x\) is not a function of \(y\).

The function is one-to-one if and only if \(y\) is a function of \(x\) and \(x\) is a function of \(y\).

Below, if the function is not one-to-one, some failed line tests are shown:
Question

Is this curve consistent with \(y\) being a function of \(x\)?

Is this curve consistent with \(x\) being a function of \(y\)?

Is this curve consistent with a one-to-one function?
Solution

It is best to visualize the vertical line test and the horizontal line test. If you can draw a vertical line through 2 (or more) points, then \(y\) is not a function of \(x\). If you can draw a horizontal line through 2 (or more) points, then \(x\) is not a function of \(y\).

The function is one-to-one if and only if \(y\) is a function of \(x\) and \(x\) is a function of \(y\).

Below, if the function is not one-to-one, some failed line tests are shown:
Question

Is this curve consistent with \(y\) being a function of \(x\)?

Is this curve consistent with \(x\) being a function of \(y\)?

Is this curve consistent with a one-to-one function?
Solution

It is best to visualize the vertical line test and the horizontal line test. If you can draw a vertical line through 2 (or more) points, then \(y\) is not a function of \(x\). If you can draw a horizontal line through 2 (or more) points, then \(x\) is not a function of \(y\).

The function is one-to-one if and only if \(y\) is a function of \(x\) and \(x\) is a function of \(y\).

Below, if the function is not one-to-one, some failed line tests are shown:
Question

Is this curve consistent with \(y\) being a function of \(x\)?

Is this curve consistent with \(x\) being a function of \(y\)?

Is this curve consistent with a one-to-one function?
Solution

It is best to visualize the vertical line test and the horizontal line test. If you can draw a vertical line through 2 (or more) points, then \(y\) is not a function of \(x\). If you can draw a horizontal line through 2 (or more) points, then \(x\) is not a function of \(y\).

The function is one-to-one if and only if \(y\) is a function of \(x\) and \(x\) is a function of \(y\).

Below, if the function is not one-to-one, some failed line tests are shown:
Question

Is this curve consistent with \(y\) being a function of \(x\)?

Is this curve consistent with \(x\) being a function of \(y\)?

Is this curve consistent with a one-to-one function?
Solution

It is best to visualize the vertical line test and the horizontal line test. If you can draw a vertical line through 2 (or more) points, then \(y\) is not a function of \(x\). If you can draw a horizontal line through 2 (or more) points, then \(x\) is not a function of \(y\).

The function is one-to-one if and only if \(y\) is a function of \(x\) and \(x\) is a function of \(y\).

Below, if the function is not one-to-one, some failed line tests are shown:
Question

Is this curve consistent with \(y\) being a function of \(x\)?

Is this curve consistent with \(x\) being a function of \(y\)?

Is this curve consistent with a one-to-one function?
Solution

It is best to visualize the vertical line test and the horizontal line test. If you can draw a vertical line through 2 (or more) points, then \(y\) is not a function of \(x\). If you can draw a horizontal line through 2 (or more) points, then \(x\) is not a function of \(y\).

The function is one-to-one if and only if \(y\) is a function of \(x\) and \(x\) is a function of \(y\).

Below, if the function is not one-to-one, some failed line tests are shown:
Question

Is this curve consistent with \(y\) being a function of \(x\)?

Is this curve consistent with \(x\) being a function of \(y\)?

Is this curve consistent with a one-to-one function?
Solution

It is best to visualize the vertical line test and the horizontal line test. If you can draw a vertical line through 2 (or more) points, then \(y\) is not a function of \(x\). If you can draw a horizontal line through 2 (or more) points, then \(x\) is not a function of \(y\).

The function is one-to-one if and only if \(y\) is a function of \(x\) and \(x\) is a function of \(y\).

Below, if the function is not one-to-one, some failed line tests are shown:
Question

Is this curve consistent with \(y\) being a function of \(x\)?

Is this curve consistent with \(x\) being a function of \(y\)?

Is this curve consistent with a one-to-one function?
Solution

It is best to visualize the vertical line test and the horizontal line test. If you can draw a vertical line through 2 (or more) points, then \(y\) is not a function of \(x\). If you can draw a horizontal line through 2 (or more) points, then \(x\) is not a function of \(y\).

The function is one-to-one if and only if \(y\) is a function of \(x\) and \(x\) is a function of \(y\).

Below, if the function is not one-to-one, some failed line tests are shown:
Question

Is this curve consistent with \(y\) being a function of \(x\)?

Is this curve consistent with \(x\) being a function of \(y\)?

Is this curve consistent with a one-to-one function?
Solution

It is best to visualize the vertical line test and the horizontal line test. If you can draw a vertical line through 2 (or more) points, then \(y\) is not a function of \(x\). If you can draw a horizontal line through 2 (or more) points, then \(x\) is not a function of \(y\).

The function is one-to-one if and only if \(y\) is a function of \(x\) and \(x\) is a function of \(y\).

Below, if the function is not one-to-one, some failed line tests are shown:
Question

Let function \(f\) be defined as follows: \[f(x) = 2 x^{2} + 5 x + 4\] Evaluate \(f(6.11)\).

You can round (or truncate) at the tenths place.

Solution

Replace all \(x\) variables with \(6.11\) in the definition of \(f\).

\[2 (6.11)^{2} + 5 (6.11) + 4\]

You’ll want to use a calculator.
Question

Let function \(f\) be defined as follows: \[f(x) = - 4 x^{2} + 4 x + 3\] Evaluate \(f(-3.29)\).

You can round (or truncate) at the tenths place.

Solution

Replace all \(x\) variables with \(-3.29\) in the definition of \(f\).

\[- 4 (-3.29)^{2} + 4 (-3.29) + 3\]

You’ll want to use a calculator.
Question

Let function \(f\) be defined as follows: \[f(x) = - 2 x^{2} - 5 x - 3\] Evaluate \(f(-6.6)\).

You can round (or truncate) at the tenths place.

Solution

Replace all \(x\) variables with \(-6.6\) in the definition of \(f\).

\[- 2 (-6.6)^{2} - 5 (-6.6) - 3\]

You’ll want to use a calculator.
Question

Let function \(f\) be defined as follows: \[f(x) = - 2 x^{2} + 3 x + 5\] Evaluate \(f(0.24)\).

You can round (or truncate) at the tenths place.

Solution

Replace all \(x\) variables with \(0.24\) in the definition of \(f\).

\[- 2 (0.24)^{2} + 3 (0.24) + 5\]

You’ll want to use a calculator.
Question

Let function \(f\) be defined as follows: \[f(x) = 3 x^{2} + 4 x - 5\] Evaluate \(f(-7.27)\).

You can round (or truncate) at the tenths place.

Solution

Replace all \(x\) variables with \(-7.27\) in the definition of \(f\).

\[3 (-7.27)^{2} + 4 (-7.27) - 5\]

You’ll want to use a calculator.
Question

Let function \(f\) be defined as follows: \[f(x) = - 5 x^{2} - 2 x - 3\] Evaluate \(f(-0.56)\).

You can round (or truncate) at the tenths place.

Solution

Replace all \(x\) variables with \(-0.56\) in the definition of \(f\).

\[- 5 (-0.56)^{2} - 2 (-0.56) - 3\]

You’ll want to use a calculator.
Question

Let function \(f\) be defined as follows: \[f(x) = - 2 x^{2} + 2 x + 4\] Evaluate \(f(9.08)\).

You can round (or truncate) at the tenths place.

Solution

Replace all \(x\) variables with \(9.08\) in the definition of \(f\).

\[- 2 (9.08)^{2} + 2 (9.08) + 4\]

You’ll want to use a calculator.
Question

Let function \(f\) be defined as follows: \[f(x) = - 3 x^{2} - 5 x + 5\] Evaluate \(f(2.29)\).

You can round (or truncate) at the tenths place.

Solution

Replace all \(x\) variables with \(2.29\) in the definition of \(f\).

\[- 3 (2.29)^{2} - 5 (2.29) + 5\]

You’ll want to use a calculator.
Question

Let function \(f\) be defined as follows: \[f(x) = - 3 x^{2} - 2 x + 4\] Evaluate \(f(-1.82)\).

You can round (or truncate) at the tenths place.

Solution

Replace all \(x\) variables with \(-1.82\) in the definition of \(f\).

\[- 3 (-1.82)^{2} - 2 (-1.82) + 4\]

You’ll want to use a calculator.
Question

Let function \(f\) be defined as follows: \[f(x) = 5 x^{2} + 4 x - 3\] Evaluate \(f(4.18)\).

You can round (or truncate) at the tenths place.

Solution

Replace all \(x\) variables with \(4.18\) in the definition of \(f\).

\[5 (4.18)^{2} + 4 (4.18) - 3\]

You’ll want to use a calculator.
Question

A function \(f\) is graphed below.
- Evaluate \(f(7)\).
Solution

Find the correct point on the graph where \(x=7\).

\[f(7)=1\]
Question

A function \(f\) is graphed below.
- Evaluate \(f(7)\).
Solution

Find the correct point on the graph where \(x=7\).

\[f(7)=6\]
Question

A function \(f\) is graphed below.
- Evaluate \(f(3)\).
Solution

Find the correct point on the graph where \(x=3\).

\[f(3)=7\]
Question

A function \(f\) is graphed below.
- Evaluate \(f(5)\).
Solution

Find the correct point on the graph where \(x=5\).

\[f(5)=9\]
Question

A function \(f\) is graphed below.
- Evaluate \(f(8)\).
Solution

Find the correct point on the graph where \(x=8\).

\[f(8)=3\]
Question

A function \(f\) is graphed below.
- Evaluate \(f(9)\).
Solution

Find the correct point on the graph where \(x=9\).

\[f(9)=5\]
Question

A function \(f\) is graphed below.
- Evaluate \(f(8)\).
Solution

Find the correct point on the graph where \(x=8\).

\[f(8)=2\]
Question

A function \(f\) is graphed below.
- Evaluate \(f(7)\).
Solution

Find the correct point on the graph where \(x=7\).

\[f(7)=9\]
Question

A function \(f\) is graphed below.
- Evaluate \(f(8)\).
Solution

Find the correct point on the graph where \(x=8\).

\[f(8)=1\]
Question

A function \(f\) is graphed below.
- Evaluate \(f(5)\).
Solution

Find the correct point on the graph where \(x=5\).

\[f(5)=8\]
Question

A function \(f\) is graphed below.
- If \(f(x)=8\), then find \(x\).
Solution

Find the correct point on the graph where \(f(x)=8\).

\[f(6)=8\]
Question

A function \(f\) is graphed below.
- If \(f(x)=7\), then find \(x\).
Solution

Find the correct point on the graph where \(f(x)=7\).

\[f(8)=7\]
Question

A function \(f\) is graphed below.
- If \(f(x)=8\), then find \(x\).
Solution

Find the correct point on the graph where \(f(x)=8\).

\[f(1)=8\]
Question

A function \(f\) is graphed below.
- If \(f(x)=4\), then find \(x\).
Solution

Find the correct point on the graph where \(f(x)=4\).

\[f(5)=4\]
Question

A function \(f\) is graphed below.
- If \(f(x)=4\), then find \(x\).
Solution

Find the correct point on the graph where \(f(x)=4\).

\[f(9)=4\]
Question

A function \(f\) is graphed below.
- If \(f(x)=7\), then find \(x\).
Solution

Find the correct point on the graph where \(f(x)=7\).

\[f(9)=7\]
Question

A function \(f\) is graphed below.
- If \(f(x)=3\), then find \(x\).
Solution

Find the correct point on the graph where \(f(x)=3\).

\[f(2)=3\]
Question

A function \(f\) is graphed below.
- If \(f(x)=4\), then find \(x\).
Solution

Find the correct point on the graph where \(f(x)=4\).

\[f(8)=4\]
Question

A function \(f\) is graphed below.
- If \(f(x)=7\), then find \(x\).
Solution

Find the correct point on the graph where \(f(x)=7\).

\[f(5)=7\]
Question

A function \(f\) is graphed below.
- If \(f(x)=4\), then find \(x\).
Solution

Find the correct point on the graph where \(f(x)=4\).

\[f(9)=4\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ - 5 x^{3} + 5 x^{2} - 2 x + 4\] Which of the following expressions is equivalent to \(f(-a)\)?
1. \(5 a^{3} + 5 a^{2} + 2 a + 4\)
2. \(5 a^{3} - 5 a^{2} + 2 a - 4\)
3. \(- 5 a^{3} + 5 a^{2} - 2 a + 4\)
4. \(- 5 a^{3} - 5 a^{2} - 2 a - 4\)
Solution

Substitute \(-a\) for \(x\).

\[f(-a) ~=~ - 5 (-a)^{3} + 5 (-a)^{2} - 2 (-a) + 4\]

Simplify. Remember, a negative raised to an even power is positive and a negative raised to an odd power is negative.

\[f(-a) ~=~ 5 a^{3} + 5 a^{2} + 2 a + 4\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ 2 x^{3} + 4 x^{2} - 2 x + 5\] Which of the following expressions is equivalent to \(f(-a)\)?
1. \(- 2 a^{3} + 4 a^{2} + 2 a + 5\)
2. \(2 a^{3} - 4 a^{2} - 2 a - 5\)
3. \(- 2 a^{3} - 4 a^{2} + 2 a - 5\)
4. \(2 a^{3} + 4 a^{2} - 2 a + 5\)
Solution

Substitute \(-a\) for \(x\).

\[f(-a) ~=~ 2 (-a)^{3} + 4 (-a)^{2} - 2 (-a) + 5\]

Simplify. Remember, a negative raised to an even power is positive and a negative raised to an odd power is negative.

\[f(-a) ~=~ - 2 a^{3} + 4 a^{2} + 2 a + 5\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ - 3 x^{3} + 4 x^{2} + 5 x + 3\] Which of the following expressions is equivalent to \(f(-a)\)?
1. \(3 a^{3} - 4 a^{2} - 5 a - 3\)
2. \(3 a^{3} + 4 a^{2} - 5 a + 3\)
3. \(- 3 a^{3} - 4 a^{2} + 5 a - 3\)
4. \(- 3 a^{3} + 4 a^{2} + 5 a + 3\)
Solution

Substitute \(-a\) for \(x\).

\[f(-a) ~=~ - 3 (-a)^{3} + 4 (-a)^{2} + 5 (-a) + 3\]

Simplify. Remember, a negative raised to an even power is positive and a negative raised to an odd power is negative.

\[f(-a) ~=~ 3 a^{3} + 4 a^{2} - 5 a + 3\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ - 2 x^{3} + 5 x^{2} + 3 x + 2\] Which of the following expressions is equivalent to \(f(-a)\)?
1. \(- 2 a^{3} + 5 a^{2} + 3 a + 2\)
2. \(2 a^{3} + 5 a^{2} - 3 a + 2\)
3. \(2 a^{3} - 5 a^{2} - 3 a - 2\)
4. \(- 2 a^{3} - 5 a^{2} + 3 a - 2\)
Solution

Substitute \(-a\) for \(x\).

\[f(-a) ~=~ - 2 (-a)^{3} + 5 (-a)^{2} + 3 (-a) + 2\]

Simplify. Remember, a negative raised to an even power is positive and a negative raised to an odd power is negative.

\[f(-a) ~=~ 2 a^{3} + 5 a^{2} - 3 a + 2\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ - 2 x^{3} - 3 x^{2} - 4 x - 5\] Which of the following expressions is equivalent to \(f(-a)\)?
1. \(2 a^{3} - 3 a^{2} + 4 a - 5\)
2. \(- 2 a^{3} - 3 a^{2} - 4 a - 5\)
3. \(2 a^{3} + 3 a^{2} + 4 a + 5\)
4. \(- 2 a^{3} + 3 a^{2} - 4 a + 5\)
Solution

Substitute \(-a\) for \(x\).

\[f(-a) ~=~ - 2 (-a)^{3} - 3 (-a)^{2} - 4 (-a) - 5\]

Simplify. Remember, a negative raised to an even power is positive and a negative raised to an odd power is negative.

\[f(-a) ~=~ 2 a^{3} - 3 a^{2} + 4 a - 5\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ 4 x^{3} - 2 x^{2} + 3 x - 5\] Which of the following expressions is equivalent to \(f(-a)\)?
1. \(- 4 a^{3} + 2 a^{2} - 3 a + 5\)
2. \(- 4 a^{3} - 2 a^{2} - 3 a - 5\)
3. \(4 a^{3} + 2 a^{2} + 3 a + 5\)
4. \(4 a^{3} - 2 a^{2} + 3 a - 5\)
Solution

Substitute \(-a\) for \(x\).

\[f(-a) ~=~ 4 (-a)^{3} - 2 (-a)^{2} + 3 (-a) - 5\]

Simplify. Remember, a negative raised to an even power is positive and a negative raised to an odd power is negative.

\[f(-a) ~=~ - 4 a^{3} - 2 a^{2} - 3 a - 5\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ - 2 x^{3} + 4 x^{2} + 2 x + 3\] Which of the following expressions is equivalent to \(f(-a)\)?
1. \(- 2 a^{3} + 4 a^{2} + 2 a + 3\)
2. \(2 a^{3} + 4 a^{2} - 2 a + 3\)
3. \(2 a^{3} - 4 a^{2} - 2 a - 3\)
4. \(- 2 a^{3} - 4 a^{2} + 2 a - 3\)
Solution

Substitute \(-a\) for \(x\).

\[f(-a) ~=~ - 2 (-a)^{3} + 4 (-a)^{2} + 2 (-a) + 3\]

Simplify. Remember, a negative raised to an even power is positive and a negative raised to an odd power is negative.

\[f(-a) ~=~ 2 a^{3} + 4 a^{2} - 2 a + 3\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ 2 x^{3} - 3 x^{2} + 5 x - 4\] Which of the following expressions is equivalent to \(f(-a)\)?
1. \(- 2 a^{3} + 3 a^{2} - 5 a + 4\)
2. \(- 2 a^{3} - 3 a^{2} - 5 a - 4\)
3. \(2 a^{3} + 3 a^{2} + 5 a + 4\)
4. \(2 a^{3} - 3 a^{2} + 5 a - 4\)
Solution

Substitute \(-a\) for \(x\).

\[f(-a) ~=~ 2 (-a)^{3} - 3 (-a)^{2} + 5 (-a) - 4\]

Simplify. Remember, a negative raised to an even power is positive and a negative raised to an odd power is negative.

\[f(-a) ~=~ - 2 a^{3} - 3 a^{2} - 5 a - 4\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ 4 x^{3} + 2 x^{2} + 3 x + 5\] Which of the following expressions is equivalent to \(f(-a)\)?
1. \(- 4 a^{3} - 2 a^{2} - 3 a - 5\)
2. \(4 a^{3} - 2 a^{2} + 3 a - 5\)
3. \(- 4 a^{3} + 2 a^{2} - 3 a + 5\)
4. \(4 a^{3} + 2 a^{2} + 3 a + 5\)
Solution

Substitute \(-a\) for \(x\).

\[f(-a) ~=~ 4 (-a)^{3} + 2 (-a)^{2} + 3 (-a) + 5\]

Simplify. Remember, a negative raised to an even power is positive and a negative raised to an odd power is negative.

\[f(-a) ~=~ - 4 a^{3} + 2 a^{2} - 3 a + 5\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ 3 x^{3} + 5 x^{2} + 4 x - 5\] Which of the following expressions is equivalent to \(f(-a)\)?
1. \(3 a^{3} + 5 a^{2} + 4 a - 5\)
2. \(- 3 a^{3} + 5 a^{2} - 4 a - 5\)
3. \(3 a^{3} - 5 a^{2} + 4 a + 5\)
4. \(- 3 a^{3} - 5 a^{2} - 4 a + 5\)
Solution

Substitute \(-a\) for \(x\).

\[f(-a) ~=~ 3 (-a)^{3} + 5 (-a)^{2} + 4 (-a) - 5\]

Simplify. Remember, a negative raised to an even power is positive and a negative raised to an odd power is negative.

\[f(-a) ~=~ - 3 a^{3} + 5 a^{2} - 4 a - 5\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ - 5 x^{3} + 2 x^{2} + 4 x - 3\] Which of the following expressions is equivalent to \(-f(a)\)?
1. \(5 a^{3} - 2 a^{2} - 4 a + 3\)
2. \(5 a^{3} + 2 a^{2} - 4 a - 3\)
3. \(- 5 a^{3} - 2 a^{2} + 4 a + 3\)
4. \(- 5 a^{3} + 2 a^{2} + 4 a - 3\)
Solution

Substitute \(a\) for \(x\) and multiply the whole expression by \((-1)\).

\[-f(a) ~=~ 5 a^{3} - 2 a^{2} - 4 a + 3\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ 5 x^{3} + 2 x^{2} + 3 x - 2\] Which of the following expressions is equivalent to \(-f(a)\)?
1. \(5 a^{3} - 2 a^{2} + 3 a + 2\)
2. \(- 5 a^{3} - 2 a^{2} - 3 a + 2\)
3. \(- 5 a^{3} + 2 a^{2} - 3 a - 2\)
4. \(5 a^{3} + 2 a^{2} + 3 a - 2\)
Solution

Substitute \(a\) for \(x\) and multiply the whole expression by \((-1)\).

\[-f(a) ~=~ - 5 a^{3} - 2 a^{2} - 3 a + 2\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ 5 x^{3} - 4 x^{2} - 5 x - 3\] Which of the following expressions is equivalent to \(-f(a)\)?
1. \(5 a^{3} - 4 a^{2} - 5 a - 3\)
2. \(- 5 a^{3} + 4 a^{2} + 5 a + 3\)
3. \(- 5 a^{3} - 4 a^{2} + 5 a - 3\)
4. \(5 a^{3} + 4 a^{2} - 5 a + 3\)
Solution

Substitute \(a\) for \(x\) and multiply the whole expression by \((-1)\).

\[-f(a) ~=~ - 5 a^{3} + 4 a^{2} + 5 a + 3\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ - 4 x^{3} + 2 x^{2} - 5 x - 3\] Which of the following expressions is equivalent to \(-f(a)\)?
1. \(- 4 a^{3} - 2 a^{2} - 5 a + 3\)
2. \(4 a^{3} + 2 a^{2} + 5 a - 3\)
3. \(4 a^{3} - 2 a^{2} + 5 a + 3\)
4. \(- 4 a^{3} + 2 a^{2} - 5 a - 3\)
Solution

Substitute \(a\) for \(x\) and multiply the whole expression by \((-1)\).

\[-f(a) ~=~ 4 a^{3} - 2 a^{2} + 5 a + 3\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ 3 x^{3} + 5 x^{2} - 3 x - 4\] Which of the following expressions is equivalent to \(-f(a)\)?
1. \(3 a^{3} + 5 a^{2} - 3 a - 4\)
2. \(- 3 a^{3} + 5 a^{2} + 3 a - 4\)
3. \(3 a^{3} - 5 a^{2} - 3 a + 4\)
4. \(- 3 a^{3} - 5 a^{2} + 3 a + 4\)
Solution

Substitute \(a\) for \(x\) and multiply the whole expression by \((-1)\).

\[-f(a) ~=~ - 3 a^{3} - 5 a^{2} + 3 a + 4\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ 5 x^{3} + 3 x^{2} - 5 x + 4\] Which of the following expressions is equivalent to \(-f(a)\)?
1. \(5 a^{3} + 3 a^{2} - 5 a + 4\)
2. \(- 5 a^{3} - 3 a^{2} + 5 a - 4\)
3. \(5 a^{3} - 3 a^{2} - 5 a - 4\)
4. \(- 5 a^{3} + 3 a^{2} + 5 a + 4\)
Solution

Substitute \(a\) for \(x\) and multiply the whole expression by \((-1)\).

\[-f(a) ~=~ - 5 a^{3} - 3 a^{2} + 5 a - 4\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ - 4 x^{3} + 2 x^{2} - 3 x + 4\] Which of the following expressions is equivalent to \(-f(a)\)?
1. \(- 4 a^{3} - 2 a^{2} - 3 a - 4\)
2. \(4 a^{3} - 2 a^{2} + 3 a - 4\)
3. \(- 4 a^{3} + 2 a^{2} - 3 a + 4\)
4. \(4 a^{3} + 2 a^{2} + 3 a + 4\)
Solution

Substitute \(a\) for \(x\) and multiply the whole expression by \((-1)\).

\[-f(a) ~=~ 4 a^{3} - 2 a^{2} + 3 a - 4\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ 3 x^{3} - 5 x^{2} + 5 x + 2\] Which of the following expressions is equivalent to \(-f(a)\)?
1. \(3 a^{3} - 5 a^{2} + 5 a + 2\)
2. \(3 a^{3} + 5 a^{2} + 5 a - 2\)
3. \(- 3 a^{3} + 5 a^{2} - 5 a - 2\)
4. \(- 3 a^{3} - 5 a^{2} - 5 a + 2\)
Solution

Substitute \(a\) for \(x\) and multiply the whole expression by \((-1)\).

\[-f(a) ~=~ - 3 a^{3} + 5 a^{2} - 5 a - 2\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ 3 x^{3} - 3 x^{2} - 2 x + 4\] Which of the following expressions is equivalent to \(-f(a)\)?
1. \(3 a^{3} + 3 a^{2} - 2 a - 4\)
2. \(- 3 a^{3} - 3 a^{2} + 2 a + 4\)
3. \(3 a^{3} - 3 a^{2} - 2 a + 4\)
4. \(- 3 a^{3} + 3 a^{2} + 2 a - 4\)
Solution

Substitute \(a\) for \(x\) and multiply the whole expression by \((-1)\).

\[-f(a) ~=~ - 3 a^{3} + 3 a^{2} + 2 a - 4\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ 4 x^{3} - 4 x^{2} + 5 x + 2\] Which of the following expressions is equivalent to \(-f(a)\)?
1. \(4 a^{3} - 4 a^{2} + 5 a + 2\)
2. \(- 4 a^{3} - 4 a^{2} - 5 a + 2\)
3. \(4 a^{3} + 4 a^{2} + 5 a - 2\)
4. \(- 4 a^{3} + 4 a^{2} - 5 a - 2\)
Solution

Substitute \(a\) for \(x\) and multiply the whole expression by \((-1)\).

\[-f(a) ~=~ - 4 a^{3} + 4 a^{2} - 5 a - 2\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ 4 x^{3} - 5 x^{2} + 2 x + 5\] Simplify \(f(-x)\).

Answer: \(x^3\) \(x^2\) \(x\)
Solution

Substitute \(-x\) for \(x\) and simplify.

\[f(-x) ~=~ - 4 x^{3} - 5 x^{2} - 2 x + 5\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ - 3 x^{3} + 3 x^{2} + 5 x + 2\] Simplify \(-f(x)\).

Answer: \(x^3\) \(x^2\) \(x\)
Solution

Multiply the whole expression by \((-1)\).

\[-f(x) ~=~ 3 x^{3} - 3 x^{2} - 5 x - 2\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ 2 x^{3} + 5 x^{2} + 4 x - 4\] Simplify \(-f(x)\).

Answer: \(x^3\) \(x^2\) \(x\)
Solution

Multiply the whole expression by \((-1)\).

\[-f(x) ~=~ - 2 x^{3} - 5 x^{2} - 4 x + 4\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ 4 x^{3} - 3 x^{2} - 5 x + 5\] Simplify \(-f(x)\).

Answer: \(x^3\) \(x^2\) \(x\)
Solution

Multiply the whole expression by \((-1)\).

\[-f(x) ~=~ - 4 x^{3} + 3 x^{2} + 5 x - 5\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ 4 x^{3} + 3 x^{2} - 3 x - 5\] Simplify \(-f(x)\).

Answer: \(x^3\) \(x^2\) \(x\)
Solution

Multiply the whole expression by \((-1)\).

\[-f(x) ~=~ - 4 x^{3} - 3 x^{2} + 3 x + 5\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ - 3 x^{3} + 3 x^{2} - 5 x + 2\] Simplify \(-f(x)\).

Answer: \(x^3\) \(x^2\) \(x\)
Solution

Multiply the whole expression by \((-1)\).

\[-f(x) ~=~ 3 x^{3} - 3 x^{2} + 5 x - 2\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ 2 x^{3} - 3 x^{2} + 5 x - 2\] Simplify \(f(-x)\).

Answer: \(x^3\) \(x^2\) \(x\)
Solution

Substitute \(-x\) for \(x\) and simplify.

\[f(-x) ~=~ - 2 x^{3} - 3 x^{2} - 5 x - 2\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ 3 x^{3} - 3 x^{2} + 2 x - 5\] Simplify \(-f(x)\).

Answer: \(x^3\) \(x^2\) \(x\)
Solution

Multiply the whole expression by \((-1)\).

\[-f(x) ~=~ - 3 x^{3} + 3 x^{2} - 2 x + 5\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ 3 x^{3} - 5 x^{2} - 2 x + 2\] Simplify \(f(-x)\).

Answer: \(x^3\) \(x^2\) \(x\)
Solution

Substitute \(-x\) for \(x\) and simplify.

\[f(-x) ~=~ - 3 x^{3} - 5 x^{2} + 2 x + 2\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ - 2 x^{3} - 3 x^{2} + 3 x - 4\] Simplify \(-f(x)\).

Answer: \(x^3\) \(x^2\) \(x\)
Solution

Multiply the whole expression by \((-1)\).

\[-f(x) ~=~ 2 x^{3} + 3 x^{2} - 3 x + 4\]
Question

This question is in regards to even and odd functions.
- A function \(f\) is even if and only if \(f(-x)=f(x)\) for all \(x\). In an even function, opposite inputs give equal outputs.
- A function \(f\) is odd if and only if \(f(-x)=-f(x)\) for all \(x\). In an odd function, opposite inputs give opposite outputs.
- A function can also be neither even nor odd.
Let function \(f\) be defined as follows: \[f(x) ~=~ 4 - 3 x^{3}\] Is the function even, odd, or neither?
1. Even
2. Odd
3. Neither
Solution

First, simplify \(f(-x)\). \[f(-x) = 3 x^{3} + 4 \]

Notice that \(f(-x)\) is not equivalent to \(f(x)\) and is not equivalent to \(-f(x)\).

The function is neither.
Question

This question is in regards to even and odd functions.
- A function \(f\) is even if and only if \(f(-x)=f(x)\) for all \(x\). In an even function, opposite inputs give equal outputs.
- A function \(f\) is odd if and only if \(f(-x)=-f(x)\) for all \(x\). In an odd function, opposite inputs give opposite outputs.
- A function can also be neither even nor odd.
Let function \(f\) be defined as follows: \[f(x) ~=~ - 2 x^{2} - 4\] Is the function even, odd, or neither?
1. Even
2. Odd
3. Neither
Solution

First, simplify \(f(-x)\). \[f(-x) = - 2 x^{2} - 4 \]

Notice that \(f(-x)=f(x)\) for all \(x\).

The function is even.
Question

This question is in regards to even and odd functions.
- A function \(f\) is even if and only if \(f(-x)=f(x)\) for all \(x\). In an even function, opposite inputs give equal outputs.
- A function \(f\) is odd if and only if \(f(-x)=-f(x)\) for all \(x\). In an odd function, opposite inputs give opposite outputs.
- A function can also be neither even nor odd.
Let function \(f\) be defined as follows: \[f(x) ~=~ 5 - 5 x^{2}\] Is the function even, odd, or neither?
1. Even
2. Odd
3. Neither
Solution

First, simplify \(f(-x)\). \[f(-x) = 5 - 5 x^{2} \]

Notice that \(f(-x)=f(x)\) for all \(x\).

The function is even.
Question

This question is in regards to even and odd functions.
- A function \(f\) is even if and only if \(f(-x)=f(x)\) for all \(x\). In an even function, opposite inputs give equal outputs.
- A function \(f\) is odd if and only if \(f(-x)=-f(x)\) for all \(x\). In an odd function, opposite inputs give opposite outputs.
- A function can also be neither even nor odd.
Let function \(f\) be defined as follows: \[f(x) ~=~ 4 x^{2} - 4 x\] Is the function even, odd, or neither?
1. Even
2. Odd
3. Neither
Solution

First, simplify \(f(-x)\). \[f(-x) = 4 x^{2} + 4 x \]

Notice that \(f(-x)\) is not equivalent to \(f(x)\) and is not equivalent to \(-f(x)\).

The function is neither.
Question

This question is in regards to even and odd functions.
- A function \(f\) is even if and only if \(f(-x)=f(x)\) for all \(x\). In an even function, opposite inputs give equal outputs.
- A function \(f\) is odd if and only if \(f(-x)=-f(x)\) for all \(x\). In an odd function, opposite inputs give opposite outputs.
- A function can also be neither even nor odd.
Let function \(f\) be defined as follows: \[f(x) ~=~ - 3 x^{3} + 4 x\] Is the function even, odd, or neither?
1. Even
2. Odd
3. Neither
Solution

First, simplify \(f(-x)\). \[f(-x) = 3 x^{3} - 4 x \]

Notice that \(f(-x)=-f(x)\) for all \(x\).

The function is odd.
Question

This question is in regards to even and odd functions.
- A function \(f\) is even if and only if \(f(-x)=f(x)\) for all \(x\). In an even function, opposite inputs give equal outputs.
- A function \(f\) is odd if and only if \(f(-x)=-f(x)\) for all \(x\). In an odd function, opposite inputs give opposite outputs.
- A function can also be neither even nor odd.
Let function \(f\) be defined as follows: \[f(x) ~=~ - 3 x^{3} - 4 x\] Is the function even, odd, or neither?
1. Even
2. Odd
3. Neither
Solution

First, simplify \(f(-x)\). \[f(-x) = 3 x^{3} + 4 x \]

Notice that \(f(-x)=-f(x)\) for all \(x\).

The function is odd.
Question

This question is in regards to even and odd functions.
- A function \(f\) is even if and only if \(f(-x)=f(x)\) for all \(x\). In an even function, opposite inputs give equal outputs.
- A function \(f\) is odd if and only if \(f(-x)=-f(x)\) for all \(x\). In an odd function, opposite inputs give opposite outputs.
- A function can also be neither even nor odd.
Let function \(f\) be defined as follows: \[f(x) ~=~ - 3 x^{3} - 4 x\] Is the function even, odd, or neither?
1. Even
2. Odd
3. Neither
Solution

First, simplify \(f(-x)\). \[f(-x) = 3 x^{3} + 4 x \]

Notice that \(f(-x)=-f(x)\) for all \(x\).

The function is odd.
Question

This question is in regards to even and odd functions.
- A function \(f\) is even if and only if \(f(-x)=f(x)\) for all \(x\). In an even function, opposite inputs give equal outputs.
- A function \(f\) is odd if and only if \(f(-x)=-f(x)\) for all \(x\). In an odd function, opposite inputs give opposite outputs.
- A function can also be neither even nor odd.
Let function \(f\) be defined as follows: \[f(x) ~=~ - 2 x^{2} - 4\] Is the function even, odd, or neither?
1. Even
2. Odd
3. Neither
Solution

First, simplify \(f(-x)\). \[f(-x) = - 2 x^{2} - 4 \]

Notice that \(f(-x)=f(x)\) for all \(x\).

The function is even.
Question

This question is in regards to even and odd functions.
- A function \(f\) is even if and only if \(f(-x)=f(x)\) for all \(x\). In an even function, opposite inputs give equal outputs.
- A function \(f\) is odd if and only if \(f(-x)=-f(x)\) for all \(x\). In an odd function, opposite inputs give opposite outputs.
- A function can also be neither even nor odd.
Let function \(f\) be defined as follows: \[f(x) ~=~ - 5 x^{3} + 4 x\] Is the function even, odd, or neither?
1. Even
2. Odd
3. Neither
Solution

First, simplify \(f(-x)\). \[f(-x) = 5 x^{3} - 4 x \]

Notice that \(f(-x)=-f(x)\) for all \(x\).

The function is odd.
Question

This question is in regards to even and odd functions.
- A function \(f\) is even if and only if \(f(-x)=f(x)\) for all \(x\). In an even function, opposite inputs give equal outputs.
- A function \(f\) is odd if and only if \(f(-x)=-f(x)\) for all \(x\). In an odd function, opposite inputs give opposite outputs.
- A function can also be neither even nor odd.
Let function \(f\) be defined as follows: \[f(x) ~=~ - 5 x^{2} - 3\] Is the function even, odd, or neither?
1. Even
2. Odd
3. Neither
Solution

First, simplify \(f(-x)\). \[f(-x) = - 5 x^{2} - 3 \]

Notice that \(f(-x)=f(x)\) for all \(x\).

The function is even.
Question

This question is in regards to even and odd functions.
- A function \(f\) is even if and only if \(f(-x)=f(x)\) for all \(x\). In an even function, opposite inputs give equal outputs. An even function has reflective symmetry over the vertical axis.
- A function \(f\) is odd if and only if \(f(-x)=-f(x)\) for all \(x\). In an odd function, opposite inputs give opposite outputs. An odd function has rotational symmetry around the origin.
- A function can also be neither even nor odd.
A function \(f\) is graphed below.

Is the function’s graph consistent with the function being even, odd, or neither?
1. Even
2. Odd
3. Neither
Solution

The graph has rotational symmetry around the origin.

The function is odd.
Question

This question is in regards to even and odd functions.
- A function \(f\) is even if and only if \(f(-x)=f(x)\) for all \(x\). In an even function, opposite inputs give equal outputs. An even function has reflective symmetry over the vertical axis.
- A function \(f\) is odd if and only if \(f(-x)=-f(x)\) for all \(x\). In an odd function, opposite inputs give opposite outputs. An odd function has rotational symmetry around the origin.
- A function can also be neither even nor odd.
A function \(f\) is graphed below.

Is the function’s graph consistent with the function being even, odd, or neither?
1. Even
2. Odd
3. Neither
Solution

The graph has neither reflective nor rotational symmetry.

The function is neither.
Question

This question is in regards to even and odd functions.
- A function \(f\) is even if and only if \(f(-x)=f(x)\) for all \(x\). In an even function, opposite inputs give equal outputs. An even function has reflective symmetry over the vertical axis.
- A function \(f\) is odd if and only if \(f(-x)=-f(x)\) for all \(x\). In an odd function, opposite inputs give opposite outputs. An odd function has rotational symmetry around the origin.
- A function can also be neither even nor odd.
A function \(f\) is graphed below.

Is the function’s graph consistent with the function being even, odd, or neither?
1. Even
2. Odd
3. Neither
Solution

The graph has reflective symmetry over the \(y\) axis.

The function is even.
Question

This question is in regards to even and odd functions.
- A function \(f\) is even if and only if \(f(-x)=f(x)\) for all \(x\). In an even function, opposite inputs give equal outputs. An even function has reflective symmetry over the vertical axis.
- A function \(f\) is odd if and only if \(f(-x)=-f(x)\) for all \(x\). In an odd function, opposite inputs give opposite outputs. An odd function has rotational symmetry around the origin.
- A function can also be neither even nor odd.
A function \(f\) is graphed below.

Is the function’s graph consistent with the function being even, odd, or neither?
1. Even
2. Odd
3. Neither
Solution

The graph has reflective symmetry over the \(y\) axis.

The function is even.
Question

This question is in regards to even and odd functions.
- A function \(f\) is even if and only if \(f(-x)=f(x)\) for all \(x\). In an even function, opposite inputs give equal outputs. An even function has reflective symmetry over the vertical axis.
- A function \(f\) is odd if and only if \(f(-x)=-f(x)\) for all \(x\). In an odd function, opposite inputs give opposite outputs. An odd function has rotational symmetry around the origin.
- A function can also be neither even nor odd.
A function \(f\) is graphed below.

Is the function’s graph consistent with the function being even, odd, or neither?
1. Even
2. Odd
3. Neither
Solution

The graph has reflective symmetry over the \(y\) axis.

The function is even.
Question

This question is in regards to even and odd functions.
- A function \(f\) is even if and only if \(f(-x)=f(x)\) for all \(x\). In an even function, opposite inputs give equal outputs. An even function has reflective symmetry over the vertical axis.
- A function \(f\) is odd if and only if \(f(-x)=-f(x)\) for all \(x\). In an odd function, opposite inputs give opposite outputs. An odd function has rotational symmetry around the origin.
- A function can also be neither even nor odd.
A function \(f\) is graphed below.

Is the function’s graph consistent with the function being even, odd, or neither?
1. Even
2. Odd
3. Neither
Solution

The graph has neither reflective nor rotational symmetry.

The function is neither.
Question

This question is in regards to even and odd functions.
- A function \(f\) is even if and only if \(f(-x)=f(x)\) for all \(x\). In an even function, opposite inputs give equal outputs. An even function has reflective symmetry over the vertical axis.
- A function \(f\) is odd if and only if \(f(-x)=-f(x)\) for all \(x\). In an odd function, opposite inputs give opposite outputs. An odd function has rotational symmetry around the origin.
- A function can also be neither even nor odd.
A function \(f\) is graphed below.

Is the function’s graph consistent with the function being even, odd, or neither?
1. Even
2. Odd
3. Neither
Solution

The graph has neither reflective nor rotational symmetry.

The function is neither.
Question

This question is in regards to even and odd functions.
- A function \(f\) is even if and only if \(f(-x)=f(x)\) for all \(x\). In an even function, opposite inputs give equal outputs. An even function has reflective symmetry over the vertical axis.
- A function \(f\) is odd if and only if \(f(-x)=-f(x)\) for all \(x\). In an odd function, opposite inputs give opposite outputs. An odd function has rotational symmetry around the origin.
- A function can also be neither even nor odd.
A function \(f\) is graphed below.

Is the function’s graph consistent with the function being even, odd, or neither?
1. Even
2. Odd
3. Neither
Solution

The graph has rotational symmetry around the origin.

The function is odd.
Question

This question is in regards to even and odd functions.
- A function \(f\) is even if and only if \(f(-x)=f(x)\) for all \(x\). In an even function, opposite inputs give equal outputs. An even function has reflective symmetry over the vertical axis.
- A function \(f\) is odd if and only if \(f(-x)=-f(x)\) for all \(x\). In an odd function, opposite inputs give opposite outputs. An odd function has rotational symmetry around the origin.
- A function can also be neither even nor odd.
A function \(f\) is graphed below.

Is the function’s graph consistent with the function being even, odd, or neither?
1. Even
2. Odd
3. Neither
Solution

The graph has neither reflective nor rotational symmetry.

The function is neither.
Question

This question is in regards to even and odd functions.
- A function \(f\) is even if and only if \(f(-x)=f(x)\) for all \(x\). In an even function, opposite inputs give equal outputs. An even function has reflective symmetry over the vertical axis.
- A function \(f\) is odd if and only if \(f(-x)=-f(x)\) for all \(x\). In an odd function, opposite inputs give opposite outputs. An odd function has rotational symmetry around the origin.
- A function can also be neither even nor odd.
A function \(f\) is graphed below.

Is the function’s graph consistent with the function being even, odd, or neither?
1. Even
2. Odd
3. Neither
Solution

The graph has rotational symmetry around the origin.

The function is odd.
Question

A function \(f\) is graphed below.

Determine the domain and range. Note: wikipedia suggests using “image” instead of “range”.

Express your answer using interval notation.

What is the domain? [, ]

What is the range? [, ]
Solution

Because the curve is continuous, the domain is from the minimum \(x\) value to the maximum \(x\) value: \([4,\,8]\), and the range is from the minimum \(y\) to the maximum \(y\): \([1,\,6]\).
Question

A function \(f\) is graphed below.

Determine the domain and range. Note: wikipedia suggests using “image” instead of “range”.

Express your answer using interval notation.

What is the domain? [, ]

What is the range? [, ]
Solution

Because the curve is continuous, the domain is from the minimum \(x\) value to the maximum \(x\) value: \([1,\,8]\), and the range is from the minimum \(y\) to the maximum \(y\): \([4,\,9]\).
Question

A function \(f\) is graphed below.

Determine the domain and range. Note: wikipedia suggests using “image” instead of “range”.

Express your answer using interval notation.

What is the domain? [, ]

What is the range? [, ]
Solution

Because the curve is continuous, the domain is from the minimum \(x\) value to the maximum \(x\) value: \([3,\,8]\), and the range is from the minimum \(y\) to the maximum \(y\): \([1,\,9]\).
Question

A function \(f\) is graphed below.

Determine the domain and range. Note: wikipedia suggests using “image” instead of “range”.

Express your answer using interval notation.

What is the domain? [, ]

What is the range? [, ]
Solution

Because the curve is continuous, the domain is from the minimum \(x\) value to the maximum \(x\) value: \([3,\,9]\), and the range is from the minimum \(y\) to the maximum \(y\): \([1,\,7]\).
Question

A function \(f\) is graphed below.

Determine the domain and range. Note: wikipedia suggests using “image” instead of “range”.

Express your answer using interval notation.

What is the domain? [, ]

What is the range? [, ]
Solution

Because the curve is continuous, the domain is from the minimum \(x\) value to the maximum \(x\) value: \([3,\,9]\), and the range is from the minimum \(y\) to the maximum \(y\): \([1,\,7]\).
Question

A function \(f\) is graphed below.

Determine the domain and range. Note: wikipedia suggests using “image” instead of “range”.

Express your answer using interval notation.

What is the domain? [, ]

What is the range? [, ]
Solution

Because the curve is continuous, the domain is from the minimum \(x\) value to the maximum \(x\) value: \([2,\,6]\), and the range is from the minimum \(y\) to the maximum \(y\): \([1,\,9]\).
Question

A function \(f\) is graphed below.

Determine the domain and range. Note: wikipedia suggests using “image” instead of “range”.

Express your answer using interval notation.

What is the domain? [, ]

What is the range? [, ]
Solution

Because the curve is continuous, the domain is from the minimum \(x\) value to the maximum \(x\) value: \([1,\,8]\), and the range is from the minimum \(y\) to the maximum \(y\): \([3,\,9]\).
Question

A function \(f\) is graphed below.

Determine the domain and range. Note: wikipedia suggests using “image” instead of “range”.

Express your answer using interval notation.

What is the domain? [, ]

What is the range? [, ]
Solution

Because the curve is continuous, the domain is from the minimum \(x\) value to the maximum \(x\) value: \([2,\,9]\), and the range is from the minimum \(y\) to the maximum \(y\): \([1,\,8]\).
Question

A function \(f\) is graphed below.

Determine the domain and range. Note: wikipedia suggests using “image” instead of “range”.

Express your answer using interval notation.

What is the domain? [, ]

What is the range? [, ]
Solution

Because the curve is continuous, the domain is from the minimum \(x\) value to the maximum \(x\) value: \([1,\,8]\), and the range is from the minimum \(y\) to the maximum \(y\): \([2,\,9]\).
Question

A function \(f\) is graphed below.

Determine the domain and range. Note: wikipedia suggests using “image” instead of “range”.

Express your answer using interval notation.

What is the domain? [, ]

What is the range? [, ]
Solution

Because the curve is continuous, the domain is from the minimum \(x\) value to the maximum \(x\) value: \([3,\,8]\), and the range is from the minimum \(y\) to the maximum \(y\): \([1,\,9]\).
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ 5 x^{2} - 2 x + 2\] Which of the following expressions is equivalent to \(f(a+h)\)?
1. \(5 a^{2} - 10 a h + 2 a + 5 h^{2} - 2 h + 2\)
2. \(5 a^{2} + 10 a h + 2 a + 5 h^{2} + 2 h + 2\)
3. \(5 a^{2} + 10 a h - 2 a + 5 h^{2} - 2 h + 2\)
4. \(5 a^{2} - 10 a h - 2 a + 5 h^{2} + 2 h + 2\)
Solution

Substitute \(a+h\) for \(x\).

\[f(a+h) ~=~ 5 (h+a)^{2} - 2 (h+a) + 2\]

Simplify. It can be helpful to know the binomial expansion coefficients from Pascal’s triangle.

\[f(a+h) ~=~ 5 a^{2} + 10 a h - 2 a + 5 h^{2} - 2 h + 2\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ 2 x^{2} - 9 x - 3\] Which of the following expressions is equivalent to \(f(a+h)\)?
1. \(2 a^{2} - 4 a h - 9 a + 2 h^{2} + 9 h - 3\)
2. \(2 a^{2} - 4 a h + 9 a + 2 h^{2} - 9 h - 3\)
3. \(2 a^{2} + 4 a h - 9 a + 2 h^{2} - 9 h - 3\)
4. \(2 a^{2} + 4 a h + 9 a + 2 h^{2} + 9 h - 3\)
Solution

Substitute \(a+h\) for \(x\).

\[f(a+h) ~=~ 2 (h+a)^{2} - 9 (h+a) - 3\]

Simplify. It can be helpful to know the binomial expansion coefficients from Pascal’s triangle.

\[f(a+h) ~=~ 2 a^{2} + 4 a h - 9 a + 2 h^{2} - 9 h - 3\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ 2 x^{2} + 5 x - 9\] Which of the following expressions is equivalent to \(f(a+h)\)?
1. \(2 a^{2} - 4 a h - 5 a + 2 h^{2} + 5 h - 9\)
2. \(2 a^{2} + 4 a h - 5 a + 2 h^{2} - 5 h - 9\)
3. \(2 a^{2} - 4 a h + 5 a + 2 h^{2} - 5 h - 9\)
4. \(2 a^{2} + 4 a h + 5 a + 2 h^{2} + 5 h - 9\)
Solution

Substitute \(a+h\) for \(x\).

\[f(a+h) ~=~ 2 (h+a)^{2} + 5 (h+a) - 9\]

Simplify. It can be helpful to know the binomial expansion coefficients from Pascal’s triangle.

\[f(a+h) ~=~ 2 a^{2} + 4 a h + 5 a + 2 h^{2} + 5 h - 9\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ 3 x^{2} + 4 x - 3\] Which of the following expressions is equivalent to \(f(a+h)\)?
1. \(3 a^{2} + 6 a h + 4 a + 3 h^{2} + 4 h - 3\)
2. \(3 a^{2} - 6 a h + 4 a + 3 h^{2} - 4 h - 3\)
3. \(3 a^{2} + 6 a h - 4 a + 3 h^{2} - 4 h - 3\)
4. \(3 a^{2} - 6 a h - 4 a + 3 h^{2} + 4 h - 3\)
Solution

Substitute \(a+h\) for \(x\).

\[f(a+h) ~=~ 3 (h+a)^{2} + 4 (h+a) - 3\]

Simplify. It can be helpful to know the binomial expansion coefficients from Pascal’s triangle.

\[f(a+h) ~=~ 3 a^{2} + 6 a h + 4 a + 3 h^{2} + 4 h - 3\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ 4 x^{2} - 2 x + 5\] Which of the following expressions is equivalent to \(f(a+h)\)?
1. \(4 a^{2} + 8 a h - 2 a + 4 h^{2} - 2 h + 5\)
2. \(4 a^{2} + 8 a h + 2 a + 4 h^{2} + 2 h + 5\)
3. \(4 a^{2} - 8 a h + 2 a + 4 h^{2} - 2 h + 5\)
4. \(4 a^{2} - 8 a h - 2 a + 4 h^{2} + 2 h + 5\)
Solution

Substitute \(a+h\) for \(x\).

\[f(a+h) ~=~ 4 (h+a)^{2} - 2 (h+a) + 5\]

Simplify. It can be helpful to know the binomial expansion coefficients from Pascal’s triangle.

\[f(a+h) ~=~ 4 a^{2} + 8 a h - 2 a + 4 h^{2} - 2 h + 5\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ 5 x^{2} + 4 x - 9\] Which of the following expressions is equivalent to \(f(a+h)\)?
1. \(5 a^{2} - 10 a h - 4 a + 5 h^{2} + 4 h - 9\)
2. \(5 a^{2} + 10 a h - 4 a + 5 h^{2} - 4 h - 9\)
3. \(5 a^{2} - 10 a h + 4 a + 5 h^{2} - 4 h - 9\)
4. \(5 a^{2} + 10 a h + 4 a + 5 h^{2} + 4 h - 9\)
Solution

Substitute \(a+h\) for \(x\).

\[f(a+h) ~=~ 5 (h+a)^{2} + 4 (h+a) - 9\]

Simplify. It can be helpful to know the binomial expansion coefficients from Pascal’s triangle.

\[f(a+h) ~=~ 5 a^{2} + 10 a h + 4 a + 5 h^{2} + 4 h - 9\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ - 9 x^{2} - 2 x + 5\] Which of the following expressions is equivalent to \(f(a+h)\)?
1. \(- 9 a^{2} + 18 a h + 2 a - 9 h^{2} - 2 h + 5\)
2. \(- 9 a^{2} - 18 a h - 2 a - 9 h^{2} - 2 h + 5\)
3. \(- 9 a^{2} - 18 a h + 2 a - 9 h^{2} + 2 h + 5\)
4. \(- 9 a^{2} + 18 a h - 2 a - 9 h^{2} + 2 h + 5\)
Solution

Substitute \(a+h\) for \(x\).

\[f(a+h) ~=~ - 9 (h+a)^{2} - 2 (h+a) + 5\]

Simplify. It can be helpful to know the binomial expansion coefficients from Pascal’s triangle.

\[f(a+h) ~=~ - 9 a^{2} - 18 a h - 2 a - 9 h^{2} - 2 h + 5\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ - 3 x^{2} - 2 x - 9\] Which of the following expressions is equivalent to \(f(a+h)\)?
1. \(- 3 a^{2} - 6 a h + 2 a - 3 h^{2} + 2 h - 9\)
2. \(- 3 a^{2} + 6 a h + 2 a - 3 h^{2} - 2 h - 9\)
3. \(- 3 a^{2} + 6 a h - 2 a - 3 h^{2} + 2 h - 9\)
4. \(- 3 a^{2} - 6 a h - 2 a - 3 h^{2} - 2 h - 9\)
Solution

Substitute \(a+h\) for \(x\).

\[f(a+h) ~=~ - 3 (h+a)^{2} - 2 (h+a) - 9\]

Simplify. It can be helpful to know the binomial expansion coefficients from Pascal’s triangle.

\[f(a+h) ~=~ - 3 a^{2} - 6 a h - 2 a - 3 h^{2} - 2 h - 9\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ 2 x^{2} - 3 x + 3\] Which of the following expressions is equivalent to \(f(a+h)\)?
1. \(2 a^{2} + 4 a h + 3 a + 2 h^{2} + 3 h + 3\)
2. \(2 a^{2} - 4 a h - 3 a + 2 h^{2} + 3 h + 3\)
3. \(2 a^{2} - 4 a h + 3 a + 2 h^{2} - 3 h + 3\)
4. \(2 a^{2} + 4 a h - 3 a + 2 h^{2} - 3 h + 3\)
Solution

Substitute \(a+h\) for \(x\).

\[f(a+h) ~=~ 2 (h+a)^{2} - 3 (h+a) + 3\]

Simplify. It can be helpful to know the binomial expansion coefficients from Pascal’s triangle.

\[f(a+h) ~=~ 2 a^{2} + 4 a h - 3 a + 2 h^{2} - 3 h + 3\]
Question

Let function \(f\) be defined as follows: \[f(x) ~=~ - 2 x^{2} - 9 x + 2\] Which of the following expressions is equivalent to \(f(a+h)\)?
1. \(- 2 a^{2} - 4 a h + 9 a - 2 h^{2} + 9 h + 2\)
2. \(- 2 a^{2} + 4 a h + 9 a - 2 h^{2} - 9 h + 2\)
3. \(- 2 a^{2} + 4 a h - 9 a - 2 h^{2} + 9 h + 2\)
4. \(- 2 a^{2} - 4 a h - 9 a - 2 h^{2} - 9 h + 2\)
Solution

Substitute \(a+h\) for \(x\).

\[f(a+h) ~=~ - 2 (h+a)^{2} - 9 (h+a) + 2\]

Simplify. It can be helpful to know the binomial expansion coefficients from Pascal’s triangle.

\[f(a+h) ~=~ - 2 a^{2} - 4 a h - 9 a - 2 h^{2} - 9 h + 2\]
Question

This question is about average rate of change. For function \(f(x)\), its average rate of change between \(x=a\) and \(x=b\) equals a quotient of differences: \[\text{ave rate of change} = \frac{f(b)-f(a)}{b-a} \]

A function \(f\) is graphed below.
- Find the average rate of change between \(x=0\) and \(x=10\).
You can round to the nearest hundredth.

Solution

\[f(0)=1\]

\[f(10)=5\]

\[\text{ave rate of change} = \frac{f(10)-f(0)}{10-0} = \frac{5-1}{10-0} = \frac{4}{10} = 0.4 \]

Notice the orange segment has length of 4 and the violet segment has length of 10. The formula for average rate of change is “rise over run”.
Question

This question is about average rate of change. For function \(f(x)\), its average rate of change between \(x=a\) and \(x=b\) equals a quotient of differences: \[\text{ave rate of change} = \frac{f(b)-f(a)}{b-a} \]

A function \(f\) is graphed below.
- Find the average rate of change between \(x=5\) and \(x=9\).
You can round to the nearest hundredth.

Solution

\[f(5)=4\]

\[f(9)=6\]

\[\text{ave rate of change} = \frac{f(9)-f(5)}{9-5} = \frac{6-4}{9-5} = \frac{2}{4} = 0.5 \]

Notice the orange segment has length of 2 and the violet segment has length of 4. The formula for average rate of change is “rise over run”.
Question

This question is about average rate of change. For function \(f(x)\), its average rate of change between \(x=a\) and \(x=b\) equals a quotient of differences: \[\text{ave rate of change} = \frac{f(b)-f(a)}{b-a} \]

A function \(f\) is graphed below.
- Find the average rate of change between \(x=4\) and \(x=9\).
You can round to the nearest hundredth.

Solution

\[f(4)=1\]

\[f(9)=3\]

\[\text{ave rate of change} = \frac{f(9)-f(4)}{9-4} = \frac{3-1}{9-4} = \frac{2}{5} = 0.4 \]

Notice the orange segment has length of 2 and the violet segment has length of 5. The formula for average rate of change is “rise over run”.
Question

This question is about average rate of change. For function \(f(x)\), its average rate of change between \(x=a\) and \(x=b\) equals a quotient of differences: \[\text{ave rate of change} = \frac{f(b)-f(a)}{b-a} \]

A function \(f\) is graphed below.
- Find the average rate of change between \(x=9\) and \(x=10\).
You can round to the nearest hundredth.

Solution

\[f(9)=1\]

\[f(10)=6\]

\[\text{ave rate of change} = \frac{f(10)-f(9)}{10-9} = \frac{6-1}{10-9} = \frac{5}{1} = 5 \]

Notice the orange segment has length of 5 and the violet segment has length of 1. The formula for average rate of change is “rise over run”.
Question

This question is about average rate of change. For function \(f(x)\), its average rate of change between \(x=a\) and \(x=b\) equals a quotient of differences: \[\text{ave rate of change} = \frac{f(b)-f(a)}{b-a} \]

A function \(f\) is graphed below.
- Find the average rate of change between \(x=2\) and \(x=4\).
You can round to the nearest hundredth.

Solution

\[f(2)=1\]

\[f(4)=9\]

\[\text{ave rate of change} = \frac{f(4)-f(2)}{4-2} = \frac{9-1}{4-2} = \frac{8}{2} = 4 \]

Notice the orange segment has length of 8 and the violet segment has length of 2. The formula for average rate of change is “rise over run”.
Question

This question is about average rate of change. For function \(f(x)\), its average rate of change between \(x=a\) and \(x=b\) equals a quotient of differences: \[\text{ave rate of change} = \frac{f(b)-f(a)}{b-a} \]

A function \(f\) is graphed below.
- Find the average rate of change between \(x=6\) and \(x=10\).
You can round to the nearest hundredth.

Solution

\[f(6)=5\]

\[f(10)=1\]

\[\text{ave rate of change} = \frac{f(10)-f(6)}{10-6} = \frac{1-5}{10-6} = \frac{-4}{4} = -1 \]

Notice the orange segment has length of 4 and the violet segment has length of 4. The formula for average rate of change is “rise over run”.
Question

This question is about average rate of change. For function \(f(x)\), its average rate of change between \(x=a\) and \(x=b\) equals a quotient of differences: \[\text{ave rate of change} = \frac{f(b)-f(a)}{b-a} \]

A function \(f\) is graphed below.
- Find the average rate of change between \(x=2\) and \(x=3\).
You can round to the nearest hundredth.

Solution

\[f(2)=4\]

\[f(3)=9\]

\[\text{ave rate of change} = \frac{f(3)-f(2)}{3-2} = \frac{9-4}{3-2} = \frac{5}{1} = 5 \]

Notice the orange segment has length of 5 and the violet segment has length of 1. The formula for average rate of change is “rise over run”.
Question

This question is about average rate of change. For function \(f(x)\), its average rate of change between \(x=a\) and \(x=b\) equals a quotient of differences: \[\text{ave rate of change} = \frac{f(b)-f(a)}{b-a} \]

A function \(f\) is graphed below.
- Find the average rate of change between \(x=5\) and \(x=10\).
You can round to the nearest hundredth.

Solution

\[f(5)=6\]

\[f(10)=2\]

\[\text{ave rate of change} = \frac{f(10)-f(5)}{10-5} = \frac{2-6}{10-5} = \frac{-4}{5} = -0.8 \]

Notice the orange segment has length of 4 and the violet segment has length of 5. The formula for average rate of change is “rise over run”.
Question

This question is about average rate of change. For function \(f(x)\), its average rate of change between \(x=a\) and \(x=b\) equals a quotient of differences: \[\text{ave rate of change} = \frac{f(b)-f(a)}{b-a} \]

A function \(f\) is graphed below.
- Find the average rate of change between \(x=5\) and \(x=7\).
You can round to the nearest hundredth.

Solution

\[f(5)=9\]

\[f(7)=2\]

\[\text{ave rate of change} = \frac{f(7)-f(5)}{7-5} = \frac{2-9}{7-5} = \frac{-7}{2} = -3.5 \]

Notice the orange segment has length of 7 and the violet segment has length of 2. The formula for average rate of change is “rise over run”.
Question

This question is about average rate of change. For function \(f(x)\), its average rate of change between \(x=a\) and \(x=b\) equals a quotient of differences: \[\text{ave rate of change} = \frac{f(b)-f(a)}{b-a} \]

A function \(f\) is graphed below.
- Find the average rate of change between \(x=7\) and \(x=10\).
You can round to the nearest hundredth.

Solution

\[f(7)=2\]

\[f(10)=9\]

\[\text{ave rate of change} = \frac{f(10)-f(7)}{10-7} = \frac{9-2}{10-7} = \frac{7}{3} = 2.33 \]

Notice the orange segment has length of 7 and the violet segment has length of 3. The formula for average rate of change is “rise over run”.
Question

We often find the average rate of change over a small interval, to approximate the instantaneous rate of change (think of a speedometer). The step size, \(h\), is the width of the interval, so we find the average rate of change of \(f(x)\) between \(a\) and \(a+h\):

\[\text{AROC} = \frac{f(a+h)-f(a)}{h}\]

Let function \(f\) be defined as follows: \[f(x) = - 2 x^{2} - 3 x + 4\] Evaluate the average rate of change of \(f(x)\) at \(a=3\) with a step size \(h=0.01\).

You can round (or truncate) at the hundredths place.

Solution

\[f(3)=-23\] \[f(3+0.01)=-23.1502\]

\[\text{AROC} = \frac{(-23.1502)-(-23)}{0.01} = -15.02 \]
Question

We often find the average rate of change over a small interval, to approximate the instantaneous rate of change (think of a speedometer). The step size, \(h\), is the width of the interval, so we find the average rate of change of \(f(x)\) between \(a\) and \(a+h\):

\[\text{AROC} = \frac{f(a+h)-f(a)}{h}\]

Let function \(f\) be defined as follows: \[f(x) = - 4 x^{2} + 2 x - 2\] Evaluate the average rate of change of \(f(x)\) at \(a=6\) with a step size \(h=0.01\).

You can round (or truncate) at the hundredths place.

Solution

\[f(6)=-134\] \[f(6+0.01)=-134.4604\]

\[\text{AROC} = \frac{(-134.4604)-(-134)}{0.01} = -46.04 \]
Question

We often find the average rate of change over a small interval, to approximate the instantaneous rate of change (think of a speedometer). The step size, \(h\), is the width of the interval, so we find the average rate of change of \(f(x)\) between \(a\) and \(a+h\):

\[\text{AROC} = \frac{f(a+h)-f(a)}{h}\]

Let function \(f\) be defined as follows: \[f(x) = 3 x^{2} - 5 x - 2\] Evaluate the average rate of change of \(f(x)\) at \(a=-2\) with a step size \(h=0.1\).

You can round (or truncate) at the hundredths place.

Solution

\[f(-2)=20\] \[f(-2+0.1)=18.33\]

\[\text{AROC} = \frac{(18.33)-(20)}{0.1} = -16.7 \]
Question

We often find the average rate of change over a small interval, to approximate the instantaneous rate of change (think of a speedometer). The step size, \(h\), is the width of the interval, so we find the average rate of change of \(f(x)\) between \(a\) and \(a+h\):

\[\text{AROC} = \frac{f(a+h)-f(a)}{h}\]

Let function \(f\) be defined as follows: \[f(x) = - 2 x^{2} - 4 x + 4\] Evaluate the average rate of change of \(f(x)\) at \(a=-2\) with a step size \(h=0.001\).

You can round (or truncate) at the hundredths place.

Solution

\[f(-2)=4\] \[f(-2+0.001)=4.003998\]

\[\text{AROC} = \frac{(4.003998)-(4)}{0.001} = 4 \]
Question

We often find the average rate of change over a small interval, to approximate the instantaneous rate of change (think of a speedometer). The step size, \(h\), is the width of the interval, so we find the average rate of change of \(f(x)\) between \(a\) and \(a+h\):

\[\text{AROC} = \frac{f(a+h)-f(a)}{h}\]

Let function \(f\) be defined as follows: \[f(x) = - 5 x^{2} - 2 x + 4\] Evaluate the average rate of change of \(f(x)\) at \(a=9\) with a step size \(h=0.1\).

You can round (or truncate) at the hundredths place.

Solution

\[f(9)=-419\] \[f(9+0.1)=-428.25\]

\[\text{AROC} = \frac{(-428.25)-(-419)}{0.1} = -92.5 \]
Question

We often find the average rate of change over a small interval, to approximate the instantaneous rate of change (think of a speedometer). The step size, \(h\), is the width of the interval, so we find the average rate of change of \(f(x)\) between \(a\) and \(a+h\):

\[\text{AROC} = \frac{f(a+h)-f(a)}{h}\]

Let function \(f\) be defined as follows: \[f(x) = - 2 x^{2} + 5 x - 4\] Evaluate the average rate of change of \(f(x)\) at \(a=-10\) with a step size \(h=0.001\).

You can round (or truncate) at the hundredths place.

Solution

\[f(-10)=-254\] \[f(-10+0.001)=-253.955002\]

\[\text{AROC} = \frac{(-253.955002)-(-254)}{0.001} = 45 \]
Question

We often find the average rate of change over a small interval, to approximate the instantaneous rate of change (think of a speedometer). The step size, \(h\), is the width of the interval, so we find the average rate of change of \(f(x)\) between \(a\) and \(a+h\):

\[\text{AROC} = \frac{f(a+h)-f(a)}{h}\]

Let function \(f\) be defined as follows: \[f(x) = 2 x^{2} + 5 x - 3\] Evaluate the average rate of change of \(f(x)\) at \(a=9\) with a step size \(h=0.01\).

You can round (or truncate) at the hundredths place.

Solution

\[f(9)=204\] \[f(9+0.01)=204.4102\]

\[\text{AROC} = \frac{(204.4102)-(204)}{0.01} = 41.02 \]
Question

We often find the average rate of change over a small interval, to approximate the instantaneous rate of change (think of a speedometer). The step size, \(h\), is the width of the interval, so we find the average rate of change of \(f(x)\) between \(a\) and \(a+h\):

\[\text{AROC} = \frac{f(a+h)-f(a)}{h}\]

Let function \(f\) be defined as follows: \[f(x) = - 5 x^{2} - 4 x - 3\] Evaluate the average rate of change of \(f(x)\) at \(a=-2\) with a step size \(h=0.001\).

You can round (or truncate) at the hundredths place.

Solution

\[f(-2)=-15\] \[f(-2+0.001)=-14.984005\]

\[\text{AROC} = \frac{(-14.984005)-(-15)}{0.001} = 15.99 \]
Question

We often find the average rate of change over a small interval, to approximate the instantaneous rate of change (think of a speedometer). The step size, \(h\), is the width of the interval, so we find the average rate of change of \(f(x)\) between \(a\) and \(a+h\):

\[\text{AROC} = \frac{f(a+h)-f(a)}{h}\]

Let function \(f\) be defined as follows: \[f(x) = - 5 x^{2} + 4 x - 3\] Evaluate the average rate of change of \(f(x)\) at \(a=5\) with a step size \(h=0.01\).

You can round (or truncate) at the hundredths place.

Solution

\[f(5)=-108\] \[f(5+0.01)=-108.4605\]

\[\text{AROC} = \frac{(-108.4605)-(-108)}{0.01} = -46.05 \]
Question

We often find the average rate of change over a small interval, to approximate the instantaneous rate of change (think of a speedometer). The step size, \(h\), is the width of the interval, so we find the average rate of change of \(f(x)\) between \(a\) and \(a+h\):

\[\text{AROC} = \frac{f(a+h)-f(a)}{h}\]

Let function \(f\) be defined as follows: \[f(x) = - 3 x^{2} - 5 x - 2\] Evaluate the average rate of change of \(f(x)\) at \(a=0\) with a step size \(h=0.001\).

You can round (or truncate) at the hundredths place.

Solution

\[f(0)=-2\] \[f(0+0.001)=-2.005003\]

\[\text{AROC} = \frac{(-2.005003)-(-2)}{0.001} = -5 \]

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